Low density parity check encoder having length of 64800 and code rate of 7/15, and low density parity check encoding method using the same

ABSTRACT

A low density parity check (LDPC) encoder, an LDPC decoder, and an LDPC encoding method are disclosed. The LDPC encoder includes first memory, second memory, and a processor. The first memory stores an LDPC codeword having a length of 64800 and a code rate of 7/15. The second memory is initialized to 0. The processor generates the LDPC codeword corresponding to information bits by performing accumulation with respect to the second memory using a sequence corresponding to a parity check matrix (PCM).

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/641,060, filed Jul. 3, 2017, which is a continuation of U.S. patentapplication Ser. No. 15/234,531 filed Aug. 11, 2016, now U.S. Pat. No.9,729,172, which is a continuation of U.S. patent application Ser. No.14/496,700, filed Sep. 25, 2014, now U.S. Pat. No. 9,450,611, whichclaims the benefit of Korean Patent Application Nos. 10-2013-0119514,10-2013-0120573, 10-2013-0149478 and 10-2014-0120015, filed Oct. 7,2013, Oct. 10, 2013, Dec. 3, 2013 and Sep. 11, 2014, respectively, whichare hereby incorporated by reference herein in their entirety.

BACKGROUND 1. Technical Field

The present disclosure relates generally to a low density parity check(LDPC) code that is used to correct errors occurring over a wirelesschannel, and, more particularly, to an LDPC code that is applicable to adigital broadcasting system.

2. Description of the Related Art

Current terrestrial television (TV) broadcasting generates co-channelinterference across an area within a distance that is three times aservice radius, and thus the same frequency cannot be reused in the areawithin the distance that is three times the service radius. An area inwhich the same frequency cannot be reused is called a white space.Spectrum efficiency significantly deteriorates due to the occurrence ofa white space.

Accordingly, there arises a need for the development of a transmissiontechnology that facilitates the elimination of a white space and thereuse of a frequency with an emphasis on reception robustness in orderto improve spectrum efficiency.

In response to this, the paper “Cloud Transmission: A New Spectrum-ReuseFriendly Digital Terrestrial Broadcasting Transmission System” publishedon September of 2012 in IEEE Transactions on Broadcasting, Vol. 58, No.3 proposes a terrestrial cloud transmission technology that facilitatesreuse, does not generate a white space, and makes the construction andoperation of a single frequency network easy.

Using this terrestrial cloud transmission technology, a broadcastingstation can transmit the same nationwide content or locally differentcontent over a single broadcasting channel. However, for this purpose, areceiver should receive one or more terrestrial cloud broadcast signalsin an area in which signals transmitted from different transmittersoverlap each other, that is, an overlap area, over a single frequencynetwork, and then should distinguish and demodulate the receivedterrestrial cloud broadcast signals. That is, the receiver shoulddemodulate one or more cloud broadcast signals in a situation in whichco-channel interference is present and the timing and frequencysynchronization between transmitted signals are not guaranteed.

Meanwhile, Korean Patent Application Publication No. 2013-0135746entitled “Low Density Parity Check Code for Terrestrial CloudTransmission” discloses an LDPC code that is optimized for terrestrialcloud transmission and exhibits excellent performance at low code rate(<0.5).

However, Korean Patent Application Publication No. 2013-0135746 isdirected to a code length completely different from an LDPC code lengthused in the DVB broadcast standard, etc., and does not teach a specificLDPC encoding method.

SUMMARY

At least one embodiment of the present invention is directed to theprovision of a new LDPC codeword having a length of 64800 and a coderate of 7/15, which is capable of being used for general purposes.

At least one embodiment of the present invention is directed to theprovision of an LDPC encoding technique that is capable of efficientlyperforming LDPC encoding using a sequence having a number of rows equalto a value that is obtained by dividing the sum of the length of thesystematic part of an LDPC codeword, that is, 30240, and the length ofthe first parity part of the LDPC codeword, that is, 1080, by 360.

In accordance with an aspect of the present invention, there is providedan LDPC encoder, including first memory configured to store an LDPCcodeword having a length of 64800 and a code rate of 7/15; second memoryconfigured to be initialized to 0; and a processor configured togenerate the LDPC codeword corresponding to information bits byperforming accumulation with respect to the second memory using asequence corresponding to a parity check matrix (PCM).

The accumulation may be performed at parity bit addresses that areupdated using the sequence corresponding to the PCM.

The LDPC codeword may include a systematic part corresponding to theinformation bits and having a length of 30240, a first parity partcorresponding to a dual diagonal matrix included in the PCM and having alength of 1080, and a second parity part corresponding to an identitymatrix included in the PCM and having a length of 33480.

The sequence may have a number of rows equal to the sum of a valueobtained by dividing a length of the systematic part, that is, 30240, bya circulant permutation matrix (CPM) size corresponding to the PCM, thatis, 360, and a value obtained by dividing a length of the first paritypart, that is, 1080, by the CPM size.

The sequence may be represented by the following Sequence Table:

Sequence Table 1st row: 460 792 1007 4580 11452 13130 26882 27020 324392nd row: 35 472 1056 7154 12700 13326 13414 16828 19102 3rd row: 45 440772 4854 7863 26945 27684 28651 31875 4th row: 744 812 892 1509 901812925 14140 21357 25106 5th row: 271 474 761 4268 6706 9609 19701 1970724870 6th row: 223 477 662 1987 9247 18376 22148 24948 27694 7th row: 44379 786 8823 12322 14666 16377 28688 29924 8th row: 104 219 562 583219665 20615 21043 22759 32180 9th row: 41 43 870 7963 13718 14136 1721630470 33428 10th row: 592 744 887 4513 6192 18116 19482 25032 34095 11throw: 456 821 1078 7162 7443 8774 15567 17243 33085 12th row: 151 666 9776946 10358 11172 18129 19777 32234 13th row: 236 793 870 2001 6805 904713877 30131 34252 14th row: 297 698 772 3449 4204 11608 22950 2607127512 15th row: 202 428 474 3205 3726 6223 7708 20214 25283 16th row:139 719 915 1447 2938 11864 15932 21748 28598 17th row: 135 853 902 323918590 20579 30578 33374 34045 18th row: 9 13 971 11834 13642 17628 2166924741 30965 19th row: 344 531 730 1880 16895 17587 21901 28620 3195720th row: 7 192 380 3168 3729 5518 6827 20372 34168 21st row: 28 521 6814313 7465 14209 21501 23364 25980 22nd row: 269 393 898 3561 11066 1198517311 26127 30309 23rd row: 42 82 707 4880 4890 9818 23340 25959 3169524th row: 189 262 707 6573 14082 22259 24230 24390 24664 25th row: 383568 573 5498 13449 13990 16904 22629 34203 26th row: 585 596 820 24402488 21956 28261 28703 29591 27th row: 755 763 795 5636 16433 2171423452 31150 34545 28th row: 23 343 669 1159 3507 13096 17978 24241 3432129th row: 316 384 944 4872 8491 18913 21085 23198 24798 30th row: 64 314765 3706 7136 8634 14227 17127 23437 31st row: 220 693 899 8791 1241713487 18335 22126 27428 32nd row: 285 794 1045 8624 8801 9547 1916721894 32657 33rd row: 386 621 1045 1634 1882 3172 13686 16027 22448 34throw: 95 622 693 2827 7098 11452 14112 18831 31308 35th row: 446 813 9287976 8935 13146 27117 27766 33111 36th row: 89 138 241 3218 9283 2045831484 31538 34216 37th row: 277 420 704 9281 12576 12788 14496 1535720585 38th row: 141 643 758 4894 10264 15144 16357 22478 26461 39th row:17 108 160 13183 15424 17939 19276 23714 26655 40th row: 109 285 6081682 20223 21791 24615 29622 31983 41st row: 123 515 622 7037 1394615292 15606 16262 23742 42nd row: 264 565 923 6460 13622 13934 2318125475 26134 43rd row: 202 548 789 8003 10993 12478 16051 25114 2757944th row: 121 450 575 5972 10062 18693 21852 23874 28031 45th row: 507560 889 12064 13316 19629 21547 25461 28732 46th row: 664 786 1043 91379294 10163 23389 31436 34297 47th row: 45 830 907 10730 16541 2123230354 30605 31847 48th row: 203 507 1060 6971 12216 13321 17861 2267129825 49th row: 369 881 952 3035 12279 12775 17682 17805 34281 50th row:683 709 1032 3787 17623 24138 26775 31432 33626 51st row: 524 792 104212249 14765 18601 25811 32422 33163 52nd row: 137 639 688 7182 816910443 22530 24597 29039 53rd row: 159 643 749 16386 17401 24135 2842933468 33469 54th row: 107 481 555 7322 13234 19344 23498 26581 3137855th row: 249 389 523 3421 10150 17616 19085 20545 32069 56th row: 395738 1045 2415 3005 3820 19541 23543 31068 57th row: 27 293 703 1717 34608326 8501 10290 32625 58th row: 126 247 515 6031 9549 10643 22067 2949034450 59th row: 331 471 1007 3020 3922 7580 23358 28620 30946 60th row:222 542 1021 3291 3652 13130 16349 33009 34348 61st row: 532 719 10385891 7528 23252 25472 31395 31774 62nd row: 145 398 774 7816 13887 1493623708 31712 33160 63rd row: 88 536 600 1239 1887 12195 13782 16726 2799864th row: 151 269 585 1445 3178 3970 15568 20358 21051 65th row: 650 819865 15567 18546 25571 32038 33350 33620 66th row: 93 469 800 6059 1040512296 17515 21354 22231 67th row: 97 206 951 6161 16376 27022 2919230190 30665 68th row: 412 549 986 5833 10583 10766 24946 28878 3193769th row: 72 604 659 5267 12227 21714 32120 33472 33974 70th row: 25 902912 1137 2975 9642 11598 25919 28278 71st row: 420 976 1055 8473 1151220198 21662 25443 30119 72nd row: 1 24 932 6426 11899 13217 13935 1654829737 73rd row: 53 618 988 6280 7267 11676 13575 15532 25787 74th row:111 739 809 8133 12717 12741 20253 20608 27850 75th row: 120 683 94314496 15162 15440 18660 27543 32404 76th row: 600 754 1055 7873 967917351 27268 33508 77th row: 344 756 1054 7102 7193 22903 24720 2788378th row: 582 1003 1046 11344 23756 27497 27977 32853 79th row: 28 429509 11106 11767 12729 13100 31792 80th row: 131 555 907 5113 10259 1030020580 23029 81st row: 406 915 977 12244 20259 26616 27899 32228 82ndrow: 46 195 224 1229 4116 10263 13608 17830 83rd row: 19 819 953 79659998 13959 30580 30754 84th row: 164 1003 1032 12920 15975 16582 2262427357 85th row: 8433 11894 13531 17675 25889 31384 86th row: 3166 38138596 10368 25104 29584 87th row: 2466 8241 12424 13376 24837 32711

The accumulation may be performed while the rows of the sequence arebeing repeatedly changed by the CPM size of the PCM.

In accordance with an aspect of the present invention, there is providedan LDPC encoding method, including initializing first memory configuredto store an LDPC codeword having a length of 64800 and a code rate of7/15 and second memory; and generating the LDPC codeword correspondingto information bits by performing accumulation with respect to thesecond memory using a sequence corresponding to a PCM.

The accumulation may be performed at parity bit addresses that areupdated using the sequence corresponding to the PCM.

The LDPC codeword may include a systematic part corresponding to theinformation bits and having a length of 30240, a first parity partcorresponding to a dual diagonal matrix included in the PCM and having alength of 1080, and a second parity part corresponding to an identitymatrix included in the PCM and having a length of 33480.

The sequence may have a number of rows equal to the sum of a valueobtained by dividing a length of the systematic part, that is, 30240, bya circulant permutation matrix (CPM) size corresponding to the PCM, thatis, 360, and a value obtained by dividing a length of the first paritypart, that is, 1080, by the CPM size.

The sequence may be represented by the above Sequence Table.

In accordance with still another aspect of the present invention, thereis provided an LDPC decoder, including a receiving unit configured toreceive an LDPC codeword encoded using a sequence corresponding to a PCMand is represented by the above Sequence Table; and a decoding unitconfigured to restore information bits from the received LDPC codewordby performing decoding corresponding to the PCM.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, features and advantages of the presentinvention will be more clearly understood from the following detaileddescription taken in conjunction with the accompanying drawings, inwhich:

FIG. 1 is a block diagram illustrating a broadcast signal transmissionand reception system according to an embodiment of the presentinvention;

FIG. 2 is an operation flowchart illustrating a broadcast signaltransmission and reception method according to an embodiment of thepresent invention;

FIG. 3 is a diagram illustrating the structure of a PCM corresponding toan LDPC code to according to an embodiment of the present invention;

FIG. 4 is a block diagram illustrating an LDPC encoder according to anembodiment of the present invention:

FIG. 5 is a block diagram illustrating an LDPC decoder according to anembodiment of the present invention;

FIG. 6 is an operation flowchart illustrating an LDPC encoding methodaccording to an embodiment of the present invention; and

FIG. 7 is a graph plotting the performance of a QC-LDPC code having alength of 64800 and a code rate of 7/15 according to an embodiment ofthe present invention against E_(b)/N_(o).

DETAILED DESCRIPTION

Embodiments of the present invention will be described in detail belowwith reference to the accompanying drawings. Repeated descriptions anddescriptions of well-known functions and configurations that have beendeemed to make the gist of the present invention unnecessarily obscurewill be omitted below. The embodiments of the present invention areintended to fully describe the present invention to persons havingordinary knowledge in the art to which the present invention pertains.Accordingly, the shapes, sizes, etc. of components in the drawings maybe exaggerated to make the description obvious.

Embodiments of the present invention will be described in detail belowwith reference to the accompanying drawings.

FIG. 1 is a block diagram illustrating a broadcast signal transmissionand reception system according to an embodiment of the presentinvention.

Referring to FIG. 1, it can be seen that a transmitter 10 and a receiver30 communicate with each other over a wireless channel 20.

The transmitter 10 generates an n-bit codeword by encoding k informationbits using an LDPC encoder 13. The codeword is modulated by themodulator 15, and is transmitted via an antenna 17. The signaltransmitted via the wireless channel 20 is received via the antenna 31of the receiver 30, and, in the receiver 30, is subjected to a processreverse to the process in the transmitter 10. That is, the received datais demodulated by a demodulator 33, and is then decoded by an LDPCdecoder 35, thereby finally restoring the information bits.

It will be apparent to those skilled in the art that the above-describedtransmission and reception processes have been described within aminimum range required for a description of the features of the presentinvention and various processes required for data transmission may beadded.

In the following, the specific processes of encoding and decoding thatare performed using an LDPC code in the LDPC encoder 13 or LDPC decoder35 and the specific configurations of encoding and decoding devices,such as the LDPC encoder 13 and the LDPC decoder 35, are described. TheLDPC encoder 13 illustrated in FIG. 1 may have a structure illustratedin FIG. 4, and the LDPC decoder 35 may have a structure illustrated inFIG. 5.

FIG. 2 is an operation flowchart illustrating a broadcast signaltransmission and reception method according to an embodiment of thepresent invention.

Referring to FIG. 2, in the broadcast signal transmission and receptionmethod according to this embodiment of the present invention, input bits(information bits) are subjected to LDPC encoding at step S210.

That is, at step S210, an n-bit codeword is generated by encoding kinformation bits using the LDPC encoder.

In this case, step S210 may be performed as in an LDPC encoding methodillustrated in FIG. 6.

Furthermore, in the broadcast signal transmission and reception method,the encoded data is modulated at step S220.

That is, at step S220, the encoded n-bit codeword is modulated using themodulator.

Furthermore, in the broadcast signal transmission and reception method,the modulated data is transmitted at step S230.

That is, at step S230, the modulated codeword is transmitted over awireless channel via the antenna.

Furthermore, in the broadcast signal transmission and reception method,the received data is demodulated at step S240.

That is, at step S240, the signal transmitted over the wireless channelis received via the antenna of the receiver, and the received data isdemodulated using the demodulator.

Furthermore, in the broadcast signal transmission and reception method,the demodulated data is subjected to LDPC decoding at step S250.

That is, at step S250, the information bits are finally restored byperforming LDPC decoding using the demodulator of the receiver.

In this case, step S250 corresponds to a process reverse to that of theLDPC encoding method illustrated in FIG. 6, and may correspond to theLDPC decoder of FIG. 5.

An LDPC code is known as a code very close to the Shannon limit for anadditive white Gaussian noise (AWGN) channel, and has the advantages ofasymptotically excellent performance and parallelizable decodingcompared to a turbo code.

Generally, an LDPC code is defined by a low-density parity check matrix(PCM) that is randomly generated. However, a randomly generated LDPCcode requires a large amount of memory to store a PCM, and requires alot of time to access memory. In order to overcome these problems, aquasi-cyclic LDPC (QC-LDPC) code has been proposed. A QC-LDPC code thatis composed of a zero matrix or a circulant permutation matrix (CPM) isdefined by a PCM that is expressed by the following Equation 1:

$\begin{matrix}{{H = \begin{bmatrix}J^{a_{11}} & J^{a_{12}} & \ldots & J^{a_{1n}} \\J^{a_{21}} & J^{a_{22}} & \ldots & J^{a_{2n}} \\\vdots & \vdots & \ddots & \vdots \\J^{a_{m\; 1}} & J^{a_{m\; 2}} & \ldots & J^{a_{mn}}\end{bmatrix}},{{{for}\mspace{14mu} a_{ij}} \in \left\{ {0,1,\ldots\mspace{14mu},{L - 1},\infty} \right\}}} & (1)\end{matrix}$

In this equation, J is a CPM having a size of L×L, and is given as thefollowing Equation 2. In the following description, L may be 360.

$\begin{matrix}{J_{L \times L} = \begin{bmatrix}0 & 1 & 0 & \ldots & 0 \\0 & 0 & 1 & \ldots & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \ldots & 1 \\1 & 0 & 0 & \ldots & 0\end{bmatrix}} & (2)\end{matrix}$

Furthermore, J^(i) is obtained by shifting an L×L identity matrix I (J⁰)to the right i (0≤i<L) times, and is an L×L zero matrix. Accordingly, inthe case of a QC-LDPC code, it is sufficient if only index exponent i isstored in order to store J^(i), and thus the amount of memory requiredto store a PCM is considerably reduced.

FIG. 3 is a diagram illustrating the structure of a PCM corresponding toan LDPC code to according to an embodiment of the present invention.

Referring to FIG. 3, the sizes of matrices A and C are g×K and(N−K−g)×(K+g), respectively, and are composed of an L×L zero matrix anda CPM, respectively. Furthermore, matrix Z is a zero matrix having asize of g×(N−K−g), matrix D is an identity matrix having a size of(N−K−g)×(N−K−g), and matrix B is a dual diagonal matrix having a size ofg×g. In this case, the matrix B may be a matrix in which all elementsexcept elements along a diagonal line and neighboring elements below thediagonal line are 0, and may be defined as the following Equation 3:

$\begin{matrix}{B_{g \times g} = \begin{bmatrix}I_{L \times L} & 0 & 0 & \ldots & 0 & 0 & 0 \\I_{L \times L} & I_{L \times L} & 0 & \ldots & 0 & 0 & 0 \\0 & I_{L \times L} & I_{L \times L} & \vdots & 0 & 0 & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\0 & 0 & 0 & \ldots & I_{L \times L} & I_{L \times L} & 0 \\0 & 0 & 0 & \ldots & 0 & I_{L \times L} & I_{L \times L}\end{bmatrix}} & (3)\end{matrix}$where I_(L×L) is an identity matrix having a size of L×L.

That is, the matrix B may be a bit-wise dual diagonal matrix, or may bea block-wise dual diagonal matrix having identity matrices as itsblocks, as indicated by Equation 3. The bit-wise dual diagonal matrix isdisclosed in detail in Korean Patent Application Publication No.2007-0058438, etc.

In particular, it will be apparent to those skilled in the art that whenthe matrix B is a bit-wise dual diagonal matrix, it is possible toperform conversion into a Quasi-cyclic form by applying row or columnpermutation to a PCM including the matrix B and having a structureillustrated in FIG. 3.

In this case, N is the length of a codeword, and K is the length ofinformation.

The present invention proposes a newly designed QC-LDPC code in whichthe code rate thereof is 7/15 and the length of a codeword is 64800, asillustrated in the following Table 1. That is, the present inventionproposes an LDPC code that is designed to receive information having alength of 30240 and generate an LDPC codeword having a length of 64800.

Table 1 illustrates the sizes of the matrices A, B, C, D and Z of theQC-LDPC code according to the present invention:

TABLE 1 Sizes Code rate Length A B C D Z 7/15 64800 1080 × 30240 1080 ×1080 33480 × 31320 33480 × 33480 1080 × 33480

The newly designed LDPC code may be represented in the form of asequence (progression), an equivalent relationship is establishedbetween the sequence and matrix (parity bit check matrix), and thesequence may be represented, as follows:

Sequence Table   1st row: 460 792 1007 4580 11452 13130 26882 2702032439 2nd row: 35 472 1056 7154 12700 13326 13414 16828 19102 3rd row:45 440 772 4854 7863 26945 27684 28651 31875 4th row: 744 812 892 15099018 12925 14140 21357 25106 5th row: 271 474 761 4268 6706 9609 1970119707 24870 6th row: 223 477 662 1987 9247 18376 22148 24948 27694 7throw: 44 379 786 8823 12322 14666 16377 28688 29924 8th row: 104 219 5625832 19665 20615 21043 22759 32180 9th row: 41 43 870 7963 13718 1413617216 30470 33428 10th row: 592 744 887 4513 6192 18116 19482 2503234095 11th row: 456 821 1078 7162 7443 8774 15567 17243 33085 12th row:151 666 977 6946 10358 11172 18129 19777 32234 13th row: 236 793 8702001 6805 9047 13877 30131 34252 14th row: 297 698 772 3449 4204 1160822950 26071 27512 15th row: 202 428 474 3205 3726 6223 7708 20214 2528316th row: 139 719 915 1447 2938 11864 15932 21748 28598 17th row: 135853 902 3239 18590 20579 30578 33374 34045 18th row: 9 13 971 1183413642 17628 21669 24741 30965 19th row: 344 531 730 1880 16895 1758721901 28620 31957 20th row: 7 192 380 3168 3729 5518 6827 20372 3416821st row: 28 521 681 4313 7465 14209 21501 23364 25980 22nd row: 269 393898 3561 11066 11985 17311 26127 30309 23rd row: 42 82 707 4880 48909818 23340 25959 31695 24th row: 189 262 707 6573 14082 22259 2423024390 24664 25th row: 383 568 573 5498 13449 13990 16904 22629 3420326th row: 585 596 820 2440 2488 21956 28261 28703 29591 27th row: 755763 795 5636 16433 21714 23452 31150 34545 28th row: 23 343 669 11593507 13096 17978 24241 34321 29th row: 316 384 944 4872 8491 18913 2108523198 24798 30th row: 64 314 765 3706 7136 8634 14227 17127 23437 31strow: 220 693 899 8791 12417 13487 18335 22126 27428 32nd row: 285 7941045 8624 8801 9547 19167 21894 32657 33rd row: 386 621 1045 1634 18823172 13686 16027 22448 34th row: 95 622 693 2827 7098 11452 14112 1883131308 35th row: 446 813 928 7976 8935 13146 27117 27766 33111 36th row:89 138 241 3218 9283 20458 31484 31538 34216 37th row: 277 420 704 928112576 12788 14496 15357 20585 38th row: 141 643 758 4894 10264 1514416357 22478 26461 39th row: 17 108 160 13183 15424 17939 19276 2371426655 40th row: 109 285 608 1682 20223 21791 24615 29622 31983 41st row:123 515 622 7037 13946 15292 15606 16262 23742 42nd row: 264 565 9236460 13622 13934 23181 25475 26134 43rd row: 202 548 789 8003 1099312478 16051 25114 27579 44th row: 121 450 575 5972 10062 18693 2185223874 28031 45th row: 507 560 889 12064 13316 19629 21547 25461 2873246th row: 664 786 1043 9137 9294 10163 23389 31436 34297 47th row: 45830 907 10730 16541 21232 30354 30605 31847 48th row: 203 507 1060 697112216 13321 17861 22671 29825 49th row: 369 881 952 3035 12279 1277517682 17805 34281 50th row: 683 709 1032 3787 17623 24138 26775 3143233626 51st row: 524 792 1042 12249 14765 18601 25811 32422 33163 52ndrow: 137 639 688 7182 8169 10443 22530 24597 29039 53rd row: 159 643 74916386 17401 24135 28429 33468 33469 54th row: 107 481 555 7322 1323419344 23498 26581 31378 55th row: 249 389 523 3421 10150 17616 1908520545 32069 56th row: 395 738 1045 2415 3005 3820 19541 23543 31068 57throw: 27 293 703 1717 3460 8326 8501 10290 32625 58th row: 126 247 5156031 9549 10643 22067 29490 34450 59th row: 331 471 1007 3020 3922 758023358 28620 30946 60th row: 222 542 1021 3291 3652 13130 16349 3300934348 61st row: 532 719 1038 5891 7528 23252 25472 31395 31774 62nd row:145 398 774 7816 13887 14936 23708 31712 33160 63rd row: 88 536 600 12391887 12195 13782 16726 27998 64th row: 151 269 585 1445 3178 3970 1556820358 21051 65th row: 650 819 865 15567 18546 25571 32038 33350 3362066th row: 93 469 800 6059 10405 12296 17515 21354 22231 67th row: 97 206951 6161 16376 27022 29192 30190 30665 68th row: 412 549 986 5833 1058310766 24946 28878 31937 69th row: 72 604 659 5267 12227 21714 3212033472 33974 70th row: 25 902 912 1137 2975 9642 11598 25919 28278 71strow: 420 976 1055 8473 11512 20198 21662 25443 30119 72nd row: 1 24 9326426 11899 13217 13935 16548 29737 73rd row: 53 618 988 6280 7267 1167613575 15532 25787 74th row: 111 739 809 8133 12717 12741 20253 2060827850 75th row: 120 683 943 14496 15162 15440 18660 27543 32404 76throw: 600 754 1055 7873 9679 17351 27268 33508 77th row: 344 756 10547102 7193 22903 24720 27883 78th row: 582 1003 1046 11344 23756 2749727977 32853 79th row: 28 429 509 11106 11767 12729 13100 31792 80th row:131 555 907 5113 10259 10300 20580 23029 81st row: 406 915 977 1224420259 26616 27899 32228 82nd row: 46 195 224 1229 4116 10263 13608 1783083rd row: 19 819 953 7965 9998 13959 30580 30754 84th row: 164 1003 103212920 15975 16582 22624 27357 85th row: 8433 11894 13531 17675 2588931384 86th row: 3166 3813 8596 10368 25104 29584 87th row: 2466 824112424 13376 24837 32711

An LDPC code that is represented in the form of a sequence is beingwidely used in the DVB standard.

According to an embodiment of the present invention, an LDPC codepresented in the form of a sequence is encoded, as follows. It isassumed that there is an information block S=(s₀, s₁, . . . , s_(K+1))having an information size K. The LDPC encoder generates a codewordA=(λ₀, λ₁, λ₂, . . . , λ_(N−1)) having a size of N=K+M₁+M₂ using theinformation block S having a size K. In this case, M₁=g, and M₂=N−K−g.Furthermore, M₁ is the size of parity bits corresponding to the dualdiagonal matrix B, and M₂ is the size of parity bits corresponding tothe identity matrix D. The encoding process is performed, as follows:

Initialization:λ_(i)=_(i) for i=0,1, . . . ,K−1p _(j)=0 for j=0,1, . . . ,M ₁ +M ₂−1  (4)

First information bit λ₀ is accumulated at parity bit addressesspecified in the 1st row of the sequence of the Sequence Table. Forexample, in an LDPC code having a length of 64800 and a code rate of7/15, an accumulation process is as follows:p ₄₆₀ =p ₄₆₀⊕λ₀ p ₇₉₂ =p ₇₉₂⊕λ₀ p ₁₀₀₇ =p ₁₀₀₇⊕λ₀ p ₄₅₈₀ =p ₄₅₈₀⊕λ₀ p₁₁₄₅₂ =p ₁₁₄₅₂⊕λ₀ p ₁₃₁₃₀ =p ₁₃₁₃₀⊕λ₀ p ₂₆₈₈₂ =p ₂₆₈₈₂⊕λ₀ p ₂₇₀₂₀ =p₂₇₀₂₀⊕λ₀ p ₃₂₄₃₉ =p ₃₂₄₃₉⊕λ₀where the addition ⊕ occurs in GF(2).

The subsequent L−1 information bits, that is, m=1, 2, . . . , L−1, areaccumulated at parity bit addresses that are calculated by the followingEquation 5:(x+m×Q ₁)mod M ₁ if x<M ₁M ₁+{(x−M ₁ +m×Q ₂)mod m ₂} if x≥M ₁  (5)where x denotes the addresses of parity bits corresponding to the firstinformation bit λ₀, that is, the addresses of the parity bits specifiedin the first row of the sequence of the Sequence Table, Q₁=M₁/L,Q₂=M₂/L, and L=360. Furthermore, Q₁ and Q₂ are defined in the followingTable 2. For example, for an LDPC code having a length of 64800 and acode rate of 7/15, M₁=1080, Q₁=3, M₂=33480, Q₂=93 and L=360, and thefollowing operations are performed on the second bit λ₁ using Equation5:p ₄₆₃ =p ₄₆₃⊕λ₁ p ₇₉₅ =p ₇₉₅⊕λ₁ p ₁₀₁₀ =p ₁₀₁₀⊕λ₁ p ₄₆₇₃ =p ₄₆₇₃⊕λ₁ p₁₁₅₄₅ =p ₁₁₅₄₅⊕λ₁ p ₁₃₂₂₃ =p ₁₃₂₂₃⊕λ₁ p ₂₆₉₇₅ =p ₂₆₉₇₅⊕λ₁ p ₂₇₁₁₃ =p₂₇₁₁₃⊕λ₁ p ₃₂₅₃₂ =p ₃₂₅₃₂⊕λ₁

Table 2 illustrates the sizes of M₁, Q₁, M₂ and Q₂ of the designedQC-LDPC code:

TABLE 2 Sizes Code rate Length M₁ M₂ Q₁ Q₂ 7/15 64800 1080 33480 3 93

The addresses of parity bit accumulators for new 360 information bitsfrom λ_(L) to λ_(2L−1) are calculated and accumulated from Equation 5using the second row of the sequence.

In a similar manner, for all groups composed of new L information bits,the addresses of parity bit accumulators are calculated and accumulatedfrom Equation 5 using new rows of the sequence.

After all the information bits from λ₀ to λ_(K−1) have been exhausted,the operations of the following Equation 6 are sequentially performedfrom i=1:p _(i) =p _(i) ⊕p _(i−1) for i=0,1, . . . ,M ₁−1  (6)

Thereafter, when a parity interleaving operation, such as that of thefollowing Equation 7, is performed, parity bits corresponding to thedual diagonal matrix B are generated:λ_(K+L·t+s) =p _(Q) ₁ _(·s+t) for 0≤s<L,0≤t<Q ₁  (7)

When the parity bits corresponding to the dual diagonal matrix B havebeen generated using K information bits λ₀, λ₁, . . . , λ_(K−1), paritybits corresponding to the identity matrix D are generated using the M₁generated parity bits λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹.

For all groups composed of L information bits from λ_(K) to λ_(K+M) ₁⁻¹, the addresses of parity bit accumulators are calculated using thenew rows (starting with a row immediately subsequent to the last rowused when the parity bits corresponding to the dual diagonal matrix Bhave been generated) of the sequence and Equation 5, and relatedoperations are performed.

When a parity interleaving operation, such as that of the followingEquation 8, is performed after all the information bits from λ_(K) toλ_(K+m) ₁ ⁻¹, have been exhausted, parity bits corresponding to theidentity matrix D are generated:λ_(K+M) ₁ _(+L·t+s) =p _(M) ₁ _(+Q) ₂ _(·s+t) for 0≤s<L,0≤t<Q ₂  (8)

FIG. 4 is a block diagram illustrating an LDPC encoder according to anembodiment of the present invention.

Referring to FIG. 4, the LDPC encoder according to this embodiment ofthe present invention includes memory 310 and 320 and a processor 330.

The memory 310 is memory that is used to store an LDPC codeword having alength of 64800 and a code rate of 7/15.

The memory 320 is memory that is initialized to 0.

The memory 310 and the memory 320 may correspond to λ₁ (i=0,1, . . . ,N−1) and p_(j) (j=0,1, . . . , M₁+M₂−1), respectively.

The memory 310 and the memory 320 may correspond to various types ofhardware for storing sets of bits, and may correspond to datastructures, such as an array, a list, a stack and a queue.

The processor 330 generates an LDPC codeword corresponding toinformation bits by performing accumulation with respect to the memory320 using a sequence corresponding to a PCM.

In this case, the accumulation may be performed at parity bit addressesthat are updated using the sequence of the above Sequence Table.

In this case, the LDPC codeword may include a systematic part λ₀, λ₁, .. . , λ_(K−1) corresponding to the information bits and having a lengthof 30240 (=K), a first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹corresponding to a dual diagonal matrix included in the PCM and having alength of 1080 (=M₁=g), and a second parity part λ_(K+M) ₁ , λ_(K+M) ₁₊₁, . . . , λ_(K+M) ₁ _(M) ₂ ⁻¹ corresponding to an identity matrixincluded in the PCM and having a length of 33480 (=M₂).

In this case, the sequence may have a number of rows equal to the sum(30240/360+1080/360=87) of a value obtained by dividing the length ofthe systematic part, that is, 30240, by a CPM size L corresponding tothe PCM, that is, 360, and a value obtained by dividing the length M₁ ofthe first parity part, that is, 1080, by 360.

As described above, the sequence may be represented by the aboveSequence Table.

In this case, the memory 320 may have a size corresponding to the sumM₁+M₂ of the length M₁ of the first parity part and the length M₂ of thesecond parity part.

In this case, the parity bit addresses may be updated based on theresults of comparing each x of the previous parity bit addressesspecified in respective rows of the sequence with the length M₁ of thefirst parity part.

That is, the parity bit addresses may be updated using Equation 5. Inthis case, x may be the previous parity bit addresses, m may be aninformation bit index that is an integer larger than 0 and smaller thanL, L may be the CPM size of the PCM, Q₁ may be M₁/L, M₁ may be the sizeof the first parity part, Q₂ may be M₂/L, and M₂ may be the size of thesecond parity part.

In this case, it may be possible to perform the accumulation whilerepeatedly changing the rows of the sequence by the CPM size L (=360) ofthe PCM, as described above.

In this case, the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹may be generated by performing parity interleaving using the memory 310and the memory 320, as described in conjunction with Equation 7.

In this case, the second parity part λ_(K+M) ₁ , λ_(K+M) ₁ ₊₁, . . . ,λ_(K+M) ₁ _(+M) ₂ ⁻¹ may be generated by performing parity interleavingusing the memory 310 and the memory 320 after generating the firstparity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹ and then performing theaccumulation using the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M)₁ ⁻¹ and the sequence, as described in conjunction with Equation 8.

FIG. 5 is a block diagram illustrating an LDPC decoder according to anembodiment of the present invention.

Referring to FIG. 5, the LDPC decoder according to this embodiment ofthe present invention may include a receiving unit 410 and a decodingunit 420.

The receiving unit 410 receives an LDPC codeword that has been encodedusing a sequence that corresponds to a PCM and is represented by theabove Sequence Table.

The decoding unit 420 restores information bits from the received LDPCcodeword by performing decoding corresponding to the PCM.

In this case, the sequence may be used to update the parity bitaddresses of the memory, and the parity bit addresses are used foraccumulation that is performed to generate parity bits corresponding tothe LDPC codeword.

In this case, the LDPC codeword may include a systematic part λ₀, λ₁, .. . , λ_(K−1) corresponding to the information bits, a first parity partλ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹ corresponding to a dual diagonalmatrix included in the PCM, and a second parity part λ_(K+M) ₁ , λ_(K+M)₁ ₊₁, . . . , λ_(K+M) ₁ _(+M) ₂ ⁻¹ corresponding to an identity matrixincluded in the PCM.

In this case, the parity bit addresses may be updated based on theresults of comparing each x of the previous parity bit addressesspecified in respective rows of the sequence with the length M₁ of thefirst parity part.

That is, the parity bit addresses may be updated using Equation 5. Inthis case, x may be the previous parity bit addresses, m may be aninformation bit index that is an integer larger than 0 and smaller thanL, L may be the CPM size of the PCM, Q₁ may be M₁/L, M₁ may be the sizeof the first parity part, Q₂ may be M₂/L, and M₂ may be the size of thesecond parity part.

FIG. 6 is an operation flowchart illustrating an LDPC encoding methodaccording to an embodiment of the present invention.

Referring to FIG. 6, the LDPC encoding method according to thisembodiment of the present invention initializes the first memory thatstores an LDPC codeword having a length of 64800 and a code rate of7/15, and second memory at step S510.

In this case, step S510 may be performed using Equation 4.

Furthermore, in the LDPC encoding method according to this embodiment ofthe present invention, an LDPC codeword corresponding to informationbits is generated by performing accumulation with respect to the secondmemory using a sequence corresponding to a PCM at step S520.

In this case, the accumulation may be performed at parity bit addressesthat are updated using the sequence corresponding to the PCM.

In this case, the LDPC codeword may include a systematic part λ₀, λ₁, .. . , λ_(K−1) corresponding to the information bits and having a lengthof 30240 (=K), a first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹corresponding to a dual diagonal matrix included in the PCM and having alength of 1080 (=M₁=g), and a second parity part λ_(K+M) ₁ , λ_(K+M) ₁₊₁, . . . , λ_(K+M) ₁ _(+M) ₂ ⁻¹ corresponding to an identity matrixincluded in the PCM and having a length of 33480 (=M₂).

In this case, the sequence may have a number of rows equal to the sum(30240/360+1080/360=87) of a value obtained by dividing the length ofthe systematic part, that is, 30240, by a CPM size L corresponding tothe PCM, that is, 360, and a value obtained by dividing the length M₁ ofthe first parity part, that is, 1080, by 360.

As described above, the sequence may be represented by the aboveSequence Table.

In this case, the parity bit addresses may be updated based on theresults of comparing each x of the previous parity bit addressesspecified in respective rows of the sequence with the length M₁ of thefirst parity part.

That is, the parity bit addresses may be updated using Equation 5. Inthis case, x may be the previous parity bit addresses, m may be aninformation bit index that is an integer larger than 0 and smaller thanL, L may be the CPM size of the PCM, Q₁ may be M₁/L, M₁ may be the sizeof the first parity part, Q₂ may be M₂/L, and M₂ may be the size of thesecond parity part.

In this case, it may be possible to perform the accumulation whilerepeatedly changing the rows of the sequence by the CPM size L (=360) ofthe PCM, as described above.

In this case, the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹may be generated by performing parity interleaving using the memory 310and the memory 320, as described in conjunction with Equation 7.

In this case, the second parity part λ_(K+M) ₁ , λ_(K+M) ₁ ₊₁, . . . ,λ_(K+M) ₁ _(+M) ₂ ⁻¹ may be generated by performing parity interleavingusing the memory 310 and the memory 320 after generating the firstparity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹ and then performing theaccumulation using the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M)₁ ⁻¹ and the sequence, as described in conjunction with Equation 8.

FIG. 7 is a graph plotting the performance of a QC-LDPC code having alength of 64800 and a code rate of 7/15 according to an embodiment ofthe present invention against E_(b)/N_(o).

The graph illustrated in FIG. 7 illustrates results that were obtainedon the assumption that a log-likelihood ratio (LLR)-based sum-productalgorithm in which binary phase shift keying (BPSK) modulation and 50rounds of repetitive decoding were performed was used for computationalexperiments. As illustrated in FIG. 7, it can be seen that the designedcode is away from the Shannon limit by about 0.6 dB at BER=10⁻⁶.

At least one embodiment of the present invention has the advantage ofproviding a new LDPC codeword having a length of 64800 and a code rateof 7/15, which is capable of being used for general purposes.

At least one embodiment of the present invention has the advantage ofproviding an LDPC encoding technique that is capable of efficientlyperforming LDPC encoding using a sequence having a number of rows equalto a value that is obtained by dividing the sum of the length of thesystematic part of an LDPC codeword, that is, 30240, and the length ofthe first parity part of the LDPC codeword, that is, 1080, by 360.

Although the specific embodiments of the present invention have beendisclosed for illustrative purposes, those skilled in the art willappreciate that various modifications, additions and substitutions arepossible without departing from the scope and spirit of the invention asdisclosed in the accompanying claims.

What is claimed is:
 1. A low density parity check (LDPC) decoder,comprising: a receiving unit configured to receive a signalcorresponding to an LDPC codeword having a length of 64800 and a coderate of 7/15, the LDPC codeword encoded using a sequence correspondingto a parity check matrix (PCM); and a decoding unit configured toperform decoding the received signal, the decoding corresponding to theparity check matrix, wherein the sequence is represented by thefollowing Sequence Table: Sequence Table   1st row: 460 792 1007 458011452 13130 26882 27020 32439 2nd row: 35 472 1056 7154 12700 1332613414 16828 19102 3rd row: 45 440 772 4854 7863 26945 27684 28651 318754th row: 744 812 892 1509 9018 12925 14140 21357 25106 5th row: 271 474761 4268 6706 9609 19701 19707 24870 6th row: 223 477 662 1987 924718376 22148 24948 27694 7th row: 44 379 786 8823 12322 14666 16377 2868829924 8th row: 104 219 562 5832 19665 20615 21043 22759 32180 9th row:41 43 870 7963 13718 14136 17216 30470 33428 10th row: 592 744 887 45136192 18116 19482 25032 34095 11th row: 456 821 1078 7162 7443 8774 1556717243 33085 12th row: 151 666 977 6946 10358 11172 18129 19777 3223413th row: 236 793 870 2001 6805 9047 13877 30131 34252 14th row: 297 698772 3449 4204 11608 22950 26071 27512 15th row: 202 428 474 3205 37266223 7708 20214 25283 16th row: 139 719 915 1447 2938 11864 15932 2174828598 17th row: 135 853 902 3239 18590 20579 30578 33374 34045 18th row:9 13 971 11834 13642 17628 21669 24741 30965 19th row: 344 531 730 188016895 17587 21901 28620 31957 20th row: 7 192 380 3168 3729 5518 682720372 34168 21st row: 28 521 681 4313 7465 14209 21501 23364 25980 22ndrow: 269 393 898 3561 11066 11985 17311 26127 30309 23rd row: 42 82 7074880 4890 9818 23340 25959 31695 24th row: 189 262 707 6573 14082 2225924230 24390 24664 25th row: 383 568 573 5498 13449 13990 16904 2262934203 26th row: 585 596 820 2440 2488 21956 28261 28703 29591 27th row:755 763 795 5636 16433 21714 23452 31150 34545 28th row: 23 343 669 11593507 13096 17978 24241 34321 29th row: 316 384 944 4872 8491 18913 2108523198 24798 30th row: 64 314 765 3706 7136 8634 14227 17127 23437 31strow: 220 693 899 8791 12417 13487 18335 22126 27428 32nd row: 285 7941045 8624 8801 9547 19167 21894 32657 33rd row: 386 621 1045 1634 18823172 13686 16027 22448 34th row: 95 622 693 2827 7098 11452 14112 1883131308 35th row: 446 813 928 7976 8935 13146 27117 27766 33111 36th row:89 138 241 3218 9283 20458 31484 31538 34216 37th row: 277 420 704 928112576 12788 14496 15357 20585 38th row: 141 643 758 4894 10264 1514416357 22478 26461 39th row: 17 108 160 13183 15424 17939 19276 2371426655 40th row: 109 285 608 1682 20223 21791 24615 29622 31983 41st row:123 515 622 7037 13946 15292 15606 16262 23742 42nd row: 264 565 9236460 13622 13934 23181 25475 26134 43rd row: 202 548 789 8003 1099312478 16051 25114 27579 44th row: 121 450 575 5972 10062 18693 2185223874 28031 45th row: 507 560 889 12064 13316 19629 21547 25461 2873246th row: 664 786 1043 9137 9294 10163 23389 31436 34297 47th row: 45830 907 10730 16541 21232 30354 30605 31847 48th row: 203 507 1060 697112216 13321 17861 22671 29825 49th row: 369 881 952 3035 12279 1277517682 17805 34281 50th row: 683 709 1032 3787 17623 24138 26775 3143233626 51st row: 524 792 1042 12249 14765 18601 25811 32422 33163 52ndrow: 137 639 688 7182 8169 10443 22530 24597 29039 53rd row: 159 643 74916386 17401 24135 28429 33468 33469 54th row: 107 481 555 7322 1323419344 23498 26581 31378 55th row: 249 389 523 3421 10150 17616 1908520545 32069 56th row: 395 738 1045 2415 3005 3820 19541 23543 31068 57throw: 27 293 703 1717 3460 8326 8501 10290 32625 58th row: 126 247 5156031 9549 10643 22067 29490 34450 59th row: 331 471 1007 3020 3922 758023358 28620 30946 60th row: 222 542 1021 3291 3652 13130 16349 3300934348 61st row: 532 719 1038 5891 7528 23252 25472 31395 31774 62nd row:145 398 774 7816 13887 14936 23708 31712 33160 63rd row: 88 536 600 12391887 12195 13782 16726 27998 64th row: 151 269 585 1445 3178 3970 1556820358 21051 65th row: 650 819 865 15567 18546 25571 32038 33350 3362066th row: 93 469 800 6059 10405 12296 17515 21354 22231 67th row: 97 206951 6161 16376 27022 29192 30190 30665 68th row: 412 549 986 5833 1058310766 24946 28878 31937 69th row: 72 604 659 5267 12227 21714 3212033472 33974 70th row: 25 902 912 1137 2975 9642 11598 25919 28278 71strow: 420 976 1055 8473 11512 20198 21662 25443 30119 72nd row: 1 24 9326426 11899 13217 13935 16548 29737 73rd row: 53 618 988 6280 7267 1167613575 15532 25787 74th row: 111 739 809 8133 12717 12741 20253 2060827850 75th row: 120 683 943 14496 15162 15440 18660 27543 32404 76throw: 600 754 1055 7873 9679 17351 27268 33508 77th row: 344 756 10547102 7193 22903 24720 27883 78th row: 582 1003 1046 11344 23756 2749727977 32853 79th row: 28 429 509 11106 11767 12729 13100 31792 80th row:131 555 907 5113 10259 10300 20580 23029 81st row: 406 915 977 1224420259 26616 27899 32228 82nd row: 46 195 224 1229 4116 10263 13608 1783083rd row: 19 819 953 7965 9998 13959 30580 30754 84th row: 164 1003 103212920 15975 16582 22624 27357 85th row: 8433 11894 13531 17675 2588931384 86th row: 3166 3813 8596 10368 25104 29584 87th row: 2466 824112424 13376 24837
 32711.


2. The LDPC decoder of claim 1, wherein the LDPC codeword comprises asystematic part corresponding to the information bits and having alength of 30240, a first parity part corresponding to a dual diagonalmatrix included in the PCM and having a length of 1080, and a secondparity part corresponding to an identity matrix included in the PCM andhaving a length of
 33480. 3. The LDPC decoder of claim 2, wherein thesequence has a number of rows equal to a sum of a value obtained bydividing a length of the systematic part, that is, 30240, by a circulantpermutation matrix (CPM) size corresponding to the PCM, that is, 360,and a value obtained by dividing a length of the first parity part, thatis, 1080, by the CPM size.
 4. The LDPC decoder of claim 1, wherein theLDPC codeword is generated by performing accumulation with respect to amemory and the accumulation is performed at parity bit addresses thatare updated using the sequence.
 5. The LDPC decoder of claim 4, whereinthe accumulation is performed while the rows of the sequence are beingrepeatedly changed by the CPM size of the PCM.
 6. A low density paritycheck (LDPC) decoding method, comprising: receiving a signalcorresponding to an LDPC codeword having a length of 64800 and a coderate of 7/15, the LDPC codeword encoded using a sequence correspondingto a parity check matrix (PCM); and decoding the received signal, thedecoding corresponding to the parity check matrix, wherein the sequenceis represented by the following Sequence Table: Sequence Table 1st row:460 792 1007 4580 11452 13130 26882 27020 32439 2nd row: 35 472 10567154 12700 13326 13414 16828 19102 3rd row: 45 440 772 4854 7863 2694527684 28651 31875 4th row: 744 812 892 1509 9018 12925 14140 21357 251065th row: 271 474 761 4268 6706 9609 19701 19707 24870 6th row: 223 477662 1987 9247 18376 22148 24948 27694 7th row: 44 379 786 8823 1232214666 16377 28688 29924 8th row: 104 219 562 5832 19665 20615 2104322759 32180 9th row: 41 43 870 7963 13718 14136 17216 30470 33428 10throw: 592 744 887 4513 6192 18116 19482 25032 34095 11th row: 456 8211078 7162 7443 8774 15567 17243 33085 12th row: 151 666 977 6946 1035811172 18129 19777 32234 13th row: 236 793 870 2001 6805 9047 13877 3013134252 14th row: 297 698 772 3449 4204 11608 22950 26071 27512 15th row:202 428 474 3205 3726 6223 7708 20214 25283 16th row: 139 719 915 14472938 11864 15932 21748 28598 17th row: 135 853 902 3239 18590 2057930578 33374 34045 18th row: 9 13 971 11834 13642 17628 21669 24741 3096519th row: 344 531 730 1880 16895 17587 21901 28620 31957 20th row: 7 192380 3168 3729 5518 6827 20372 34168 21st row: 28 521 681 4313 7465 1420921501 23364 25980 22nd row: 269 393 898 3561 11066 11985 17311 2612730309 23rd row: 42 82 707 4880 4890 9818 23340 25959 31695 24th row: 189262 707 6573 14082 22259 24230 24390 24664 25th row: 383 568 573 549813449 13990 16904 22629 34203 26th row: 585 596 820 2440 2488 2195628261 28703 29591 27th row: 755 763 795 5636 16433 21714 23452 3115034545 28th row: 23 343 669 1159 3507 13096 17978 24241 34321 29th row:316 384 944 4872 8491 18913 21085 23198 24798 30th row: 64 314 765 37067136 8634 14227 17127 23437 31st row: 220 693 899 8791 12417 13487 1833522126 27428 32nd row: 285 794 1045 8624 8801 9547 19167 21894 32657 33rdrow: 386 621 1045 1634 1882 3172 13686 16027 22448 34th row: 95 622 6932827 7098 11452 14112 18831 31308 35th row: 446 813 928 7976 8935 1314627117 27766 33111 36th row: 89 138 241 3218 9283 20458 31484 31538 3421637th row: 277 420 704 9281 12576 12788 14496 15357 20585 38th row: 141643 758 4894 10264 15144 16357 22478 26461 39th row: 17 108 160 1318315424 17939 19276 23714 26655 40th row: 109 285 608 1682 20223 2179124615 29622 31983 41st row: 123 515 622 7037 13946 15292 15606 1626223742 42nd row: 264 565 923 6460 13622 13934 23181 25475 26134 43rd row:202 548 789 8003 10993 12478 16051 25114 27579 44th row: 121 450 5755972 10062 18693 21852 23874 28031 45th row: 507 560 889 12064 1331619629 21547 25461 28732 46th row: 664 786 1043 9137 9294 10163 2338931436 34297 47th row: 45 830 907 10730 16541 21232 30354 30605 3184748th row: 203 507 1060 6971 12216 13321 17861 22671 29825 49th row: 369881 952 3035 12279 12775 17682 17805 34281 50th row: 683 709 1032 378717623 24138 26775 31432 33626 51st row: 524 792 1042 12249 14765 1860125811 32422 33163 52nd row: 137 639 688 7182 8169 10443 22530 2459729039 53rd row: 159 643 749 16386 17401 24135 28429 33468 33469 54throw: 107 481 555 7322 13234 19344 23498 26581 31378 55th row: 249 389523 3421 10150 17616 19085 20545 32069 56th row: 395 738 1045 2415 30053820 19541 23543 31068 57th row: 27 293 703 1717 3460 8326 8501 1029032625 58th row: 126 247 515 6031 9549 10643 22067 29490 34450 59th row:331 471 1007 3020 3922 7580 23358 28620 30946 60th row: 222 542 10213291 3652 13130 16349 33009 34348 61st row: 532 719 1038 5891 7528 2325225472 31395 31774 62nd row: 145 398 774 7816 13887 14936 23708 3171233160 63rd row: 88 536 600 1239 1887 12195 13782 16726 27998 64th row:151 269 585 1445 3178 3970 15568 20358 21051 65th row: 650 819 865 1556718546 25571 32038 33350 33620 66th row: 93 469 800 6059 10405 1229617515 21354 22231 67th row: 97 206 951 6161 16376 27022 29192 3019030665 68th row: 412 549 986 5833 10583 10766 24946 28878 31937 69th row:72 604 659 5267 12227 21714 32120 33472 33974 70th row: 25 902 912 11372975 9642 11598 25919 28278 71st row: 420 976 1055 8473 11512 2019821662 25443 30119 72nd row: 1 24 932 6426 11899 13217 13935 16548 2973773rd row: 53 618 988 6280 7267 11676 13575 15532 25787 74th row: 111 739809 8133 12717 12741 20253 20608 27850 75th row: 120 683 943 14496 1516215440 18660 27543 32404 76th row: 600 754 1055 7873 9679 17351 2726833508 77th row: 344 756 1054 7102 7193 22903 24720 27883 78th row: 5821003 1046 11344 23756 27497 27977 32853 79th row: 28 429 509 11106 1176712729 13100 31792 80th row: 131 555 907 5113 10259 10300 20580 2302981st row: 406 915 977 12244 20259 26616 27899 32228 82nd row: 46 195 2241229 4116 10263 13608 17830 83rd row: 19 819 953 7965 9998 13959 3058030754 84th row: 164 1003 1032 12920 15975 16582 22624 27357 85th row:8433 11894 13531 17675 25889 31384 86th row: 3166 3813 8596 10368 2510429584 87th row: 2466 8241 12424 13376 24837 32711.